If the half-life period of a radioactive isotope is \(10 \mathrm{~s}\), then its average life will be: (a) \(14.4 \mathrm{~s}\) (b) \(1.44 \mathrm{~s}\) (c) \(0.144 \mathrm{~s}\) (d) \(2.44 \mathrm{~s}\)

Radioactive isotopes undergo a natural process known as radioactive decay, where the nucleus of an atom changes into another nucleus or into a lower energy state by emitting particles or radiation. The speed at which these isotopes decay is described in terms of their half-life, which is the time required for half the atoms in a sample to decay.

However, the average life, also termed as the mean lifetime, of a radioactive isotope delivers a different insight. It represents the average time an atom is expected to exist before it decays. This concept is vital in fields like nuclear physics and chemistry because it assists in determining the longevity and stability of isotopes.

The mathematical relationship between half-life (\( T_{1/2} \)) and average life (\( \tau \) is defined by the equation \( \tau = \frac{T_{1/2}}{\ln(2)} \) where \( \) is the natural logarithm. The natural logarithm of 2, approximately 0.693, serves as a constant conversion factor in this equation. Understanding this concept aids students in comprehending how long a radioactive isotope can be expected to last before decay significantly reduces its presence.

When solving problems involving radioactive decay, calculations are essential to predict the behavior and rate of decay of an isotope. The half-life formula is foundational, but for a comprehensive analysis, scientists often need to determine the average life, total activity, and remaining quantity of an isotope after a certain period.

To calculate the average life of an isotope, we use the equation provided in the solution step, incorporating the isotope's known half-life. Performing this calculation allows for the computation of the time you would expect a single atom of the isotope to exist before it decays.

Let's apply this to our exercise problem: by inserting the given half-life value of 10 seconds into the average life equation, we can compute its average life. This practical application of the concept ensures that students have the tools to draw connections between theoretical concepts and real-world situations, enhancing their learning and retention.

Natural logarithms play a fundamental role in several equations in chemistry, including those used in the context of radioactive decay. The natural logarithm, often denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is Euler's number (approximately 2.71828). It arises naturally in many areas of science and mathematics as it is tied to the rate of growth processes.

In radioactive decay calculations, the natural logarithm helps to simplify the expression of time-dependent decay equations. The term \( \ln(2) \) specifically appears in the formula that links an isotope's half-life to its average life. It acts as a universal bridge applying to all decaying isotopes.

Educational exercises often require students to work with natural logarithms to reinforce an understanding of exponential growth and decay models. Becoming comfortable with natural logarithms not only assists with radioactive decay problems but also empowers students tackling a vast array of scientific problems, fostering a deeper engagement with the language of science.